{"id":86,"date":"2024-02-03T11:55:17","date_gmt":"2024-02-03T03:55:17","guid":{"rendered":"https:\/\/www.nenuacm.top\/?p=86"},"modified":"2024-02-16T01:18:47","modified_gmt":"2024-02-15T17:18:47","slug":"2024-1-31-dp%e5%8a%a8%e6%80%81%e8%a7%84%e5%88%92%e4%b9%a0%e9%a2%98","status":"publish","type":"post","link":"https:\/\/www.nenuacm.top\/index.php\/2024\/02\/03\/2024-1-31-dp%e5%8a%a8%e6%80%81%e8%a7%84%e5%88%92%e4%b9%a0%e9%a2%98\/","title":{"rendered":"2024.1.31 DP\u52a8\u6001\u89c4\u5212\u4e60\u9898"},"content":{"rendered":"\n<h1 class=\"wp-block-heading\">A &#8211; Frog 2<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Problem Statement<\/h3>\n\n\n\n<p>\u73b0\u6709 <strong>N<\/strong> \u4e2a\u77f3\u5934\uff0c\u5206\u522b\u6807\u8bb0\u4e3a <strong>1, 2, &#8230;, N<\/strong>\u3002\u5bf9\u4e8e\u6bcf\u4e00\u4e2a\u77f3\u5934 <strong>i<\/strong> (<strong>1 \u2264 i \u2264 N<\/strong>)\u6709\u4e00\u4e2a\u9ad8\u5ea6\u8bb0\u4e3a <strong>hi<\/strong>\u3002<\/p>\n\n\n\n<p>\u73b0\u6709\u4e00\u53ea\u9752\u86d9\u5728\u77f3\u5934 <strong>1<\/strong> \u4e0a\uff0c\u5b83\u60f3\u901a\u8fc7\u4e00\u4e9b\u8df3\u8dc3\u5230\u8fbe\u77f3\u5934 <strong>N<\/strong>\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c\u9752\u86d9\u6b64\u65f6\u5728\u77f3\u5934 <strong>i<\/strong> \u4e0a\uff0c\u90a3\u4e48\u5b83\u53ef\u4ee5\u8df3\u5230\u77f3\u5934 <strong>i + 1<\/strong>\uff0c \u77f3\u5934 <strong>i + 2<\/strong>\uff0c &#8230; \uff0c\u77f3\u5934 <strong>i + K<\/strong> \u4e2d\u7684\u67d0\u4e00\u4e2a\u4e0a\u3002\u8fd9\u6b21\u8df3\u8dc3\u9700\u8981\u6d88\u8017\u5b83 <strong>|hi &#8211; hj|<\/strong> \u80fd\u91cf\uff0c\u5176\u4e2d <strong>j<\/strong> \u4e3a\u9752\u86d9\u8df3\u8dc3\u5230\u8fbe\u7684\u77f3\u5934\u5e8f\u53f7\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u627e\u5230\u9752\u86d9\u5230\u8fbe\u77f3\u5934 <strong>N<\/strong> \u6d88\u8017\u80fd\u91cf\u53ef\u80fd\u7684\u6700\u5c0f\u503c\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Constraints<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8f93\u5165\u7684\u6240\u6709\u503c\u5747\u4e3a\u6574\u6570\u3002<\/li>\n\n\n\n<li><strong>2 \u2264 N \u2264 105<\/strong><\/li>\n\n\n\n<li><strong>1 \u2264 K \u2264 100<\/strong><\/li>\n\n\n\n<li><strong>1 \u2264 hi \u2264 104<\/strong><\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Input<\/h3>\n\n\n\n<p>\u6807\u51c6\u8f93\u5165\u683c\u5f0f\u5982\u4e0b\uff1aN K<br>h1 h2 &#8230; hN<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Output<\/h3>\n\n\n\n<p>\u8f93\u51fa\u9752\u86d9\u6d88\u8017\u80fd\u91cf\u53ef\u80fd\u7684\u6700\u5c0f\u503c\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/A\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u8fd9\u9053\u9898\u5982\u679c\u7528\u641c\u7d22\u53bb\u505a\u7684\u8bdd\uff0c\u4e00\u4e2a\u662f\u6570\u636e\u91cf\u592a\u5927\uff0cDFS\u5f88\u53ef\u80fd\u4f1a\u7206\u6808\uff0c\u5e76\u4e14\u4e5f\u4f1a\u56e0\u4e3a\u5c42\u6570\u592a\u591a\uff0c\u5bfc\u81f4BFS\u8fc7\u6162\uff0c\u53e6\u4e00\u65b9\u9762\u662f\u641c\u7d22\u7684\u7a7a\u95f4\u662f\u4e0d\u786e\u5b9a\u7684\uff0c\u56e0\u4e3a\u6709\u4e2ak\u5728\u5f71\u54cd\uff0c\u6240\u4ee5\u641c\u7d22\u4f1a\u7279\u522b\u96be\u5199\u3002\u6240\u4ee5\u7528\u9012\u63a8\u7684\u8bdd\uff0c\u5c31\u80fd\u4ee5\u8f83\u5feb\u7684\u65f6\u95f4\u7ed3\u675f\u3002\u6211\u4eec\u53ef\u4ee5\u4ee5\u641c\u7d22\u7684\u601d\u8def\u53bb\u60f3\u8fd9\u4e2a\u72b6\u6001\u8f6c\u79fb\uff0c\u6bd4\u5982\u6211\u4eec\u5047\u8bbe\u7b2c i \u4e2a\u77f3\u5934\u5904\u9752\u86d9\u7684\u6700\u4f4e\u80fd\u8017\u662f dp[ i ] \uff0c\u90a3\u6211\u4eec\u5c31\u4f1a\u53bb\u641c\u7d22 dp[ i &#8211; 1 ] \u2026\u2026 dp[ i &#8211; k ]\uff0c\u6211\u4eec\u5c31\u53ef\u4ee5\u8bf4\uff0cdp[ i ] \u7684\u503c\u662f\u4ece dp[ i &#8211; 1 ] \u2026\u2026 dp[ i &#8211; k ] \u8f6c\u79fb\u800c\u6765\u7684\u3002<\/p>\n\n\n\n<p>\u6839\u636e\u9898\u610f\u53ef\u4ee5\u5199\u51fa\u9012\u63a8\u5f0f\uff1a<\/p>\n\n\n\n<p>$$ dp[i] = dp[i &#8211; 1] + |h_i &#8211; h_{i &#8211; 1}| \\quad \\text{\u6216\u8005} \\quad dp[i] = dp[i &#8211; 2] + |h_i &#8211; h_{i &#8211; 2}| \\quad \\text{\u7b49\u7b49\u7b49\u2026\u2026} $$<\/p>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u603b\u7ed3\u51fa\u4e00\u4e2a\u8f6c\u79fb\u5f0f\u5c31\u662f\uff1a<\/p>\n\n\n\n<p>$$ dp[i] = \\min(dp[j] + |h_i &#8211; h_j|) \\quad (1 \\leq j &lt; i) $$<\/p>\n\n\n\n<p>\u7136\u540e\u6309\u7167\u8fd9\u4e2a\u601d\u8def\u53bb\u5199\u5c31\u597d\u4e86\uff0c\u6ce8\u610f\u4e0d\u8981\u6570\u7ec4\u8d8a\u754c\uff01<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 1e5 + 7;\nint h&#91;MAXN];\nint dp&#91;MAXN];\nint main(){\n\tint n,k;\n\tcin &gt;&gt; n &gt;&gt; k;\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tcin &gt;&gt; h&#91;i];\n\t}\n\t\/\/\u8003\u8651\u8f6c\u79fb\u72b6\u6001,dp&#91;i] = min(dp&#91;j] + |hi - hj|)\n\tmemset(dp,0x3f,sizeof(dp));\n\tdp&#91;1] = 0;\n\tfor(int i = 2;i &lt;= n;i++){\n\t\tfor(int j = max(1,i - k);j &lt; i;j++){\n\t\t\tdp&#91;i] = min(dp&#91;i],dp&#91;j] + abs(h&#91;i] - h&#91;j]));\n\t\t}\n\t}\n\tcout &lt;&lt; dp&#91;n] &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">B &#8211; Vacation<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">\u95ee\u9898\u63cf\u8ff0<\/h3>\n\n\n\n<p>\u5047\u8bbe\u5c0f\u674e\u5728\u63a5\u4e0b\u6765\u7684$n$\u5929\u91cc\u53ea\u80fd\u73a9\u4e09\u79cd\u6e38\u620f\u4e4b\u4e00\uff0c\u5373\u5728\u7b2c$i$\u5929$(1 \\leq i \\leq n)$\uff0c\u4ed6\u53ef\u80fd\u4f1a\u73a9\uff1a<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u5927\u519c\u7279\u519c\uff0c\u5e76\u83b7\u5f97$a_i$\u70b9\u5feb\u4e50\u503c\u3002<\/li>\n\n\n\n<li>\u5927\u539f\u7279\u539f\uff0c\u5e76\u83b7\u5f97$b_i$\u70b9\u5feb\u4e50\u503c\u3002<\/li>\n\n\n\n<li>\u5927\u8f68\u7279\u8f68\uff0c\u5e76\u83b7\u5f97$c_i$\u70b9\u5feb\u4e50\u503c\u3002<\/li>\n<\/ol>\n\n\n\n<p>\u4f46\u5c0f\u674e\u4e0d\u4f1a\u8fde\u7eed\u73a9\u540c\u4e00\u79cd\u6e38\u620f\u4e24\u5929\u53ca\u4ee5\u4e0a\u3002\u73b0\u5728\u9700\u8981\u6c42\u51fa\u5c0f\u674e\u5728$n$\u5929\u540e\u80fd\u83b7\u5f97\u7684\u6700\u5927\u5feb\u4e50\u503c\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\"><strong>\u8f93\u5165\u683c\u5f0f<\/strong><\/h3>\n\n\n\n<p>\u8f93\u5165\u7b2c\u4e00\u884c\u5305\u542b\u4e00\u4e2a\u6574\u6570$n$\uff0c\u8868\u793a\u5929\u6570\u3002<\/p>\n\n\n\n<p>\u63a5\u4e0b\u6765$n$\u884c\uff0c\u6bcf\u884c\u5305\u542b3\u4e2a\u6574\u6570\uff0c\u5206\u522b\u8868\u793a$a_i$\uff0c$b_i$\uff0c$c_i$\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u6570\u636e\u8303\u56f4<\/h3>\n\n\n\n<p>$1 \\leq n \\leq 10^5$<\/p>\n\n\n\n<p>$1 \\leq a_i, b_i, c_i \\leq 10^4$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">\u8f93\u51fa\u683c\u5f0f<\/h3>\n\n\n\n<p>\u8f93\u51fa\u5305\u542b\u4e00\u4e2a\u6574\u6570\uff0c\u8868\u793a\u7b54\u6848\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/B\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u5bf9\u4e8e\u8fd9\u79cd\u5927\u6570\u636e\u91cf\u7684\u9898\u76ee\uff0c\u90fd\u51e0\u4e4e\u4e0d\u8003\u8651\u641c\u7d22\uff0c\u800c\u662f\u8003\u8651DP\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u4f1a\u53d1\u73b0\u8fd9\u9053\u9898\u548c\u4e0a\u4e00\u9898\u4e0d\u540c\u7684\u5730\u65b9\u5728\u4e8e\uff0c\u8fd9\u9053\u9898\u7684\u72b6\u6001\u5f88\u591a\u3002\u6211\u4eec\u6bcf\u5929\u53ef\u4ee5\u9009\u62e9\u7b2c\u4e00\u79cd\u7b2c\u4e8c\u79cd\u7b2c\u4e09\u79cd\uff0c\u4e0d\u540c\u7684\u51b3\u7b56\u53c8\u4f1a\u6709\u4e0d\u540c\u7684\u5feb\u4e50\u503c\u3002\u6211\u4eec\u53ef\u4ee5\u5148\u5047\u5b9a\u7b2c i \u5929\u6211\u4eec\u9009\u62e9\u4e86\u7b2c\u4e00\u79cd\uff0c\u90a3\u6b64\u65f6 dp[ i ] = max\uff08\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e8c\u79cd\uff0c\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e09\u79cd\uff09 + a[ i ]\uff1b\u5047\u5b9a\u7b2c i \u5929\u6211\u4eec\u9009\u62e9\u4e86\u7b2c\u4e8c\u79cd\uff0c\u90a3\u6b64\u65f6 dp[ i ] = max\uff08\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e00\u79cd\uff0c\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e09\u79cd\uff09 + b[ i ]\uff1b\u5047\u5b9a\u7b2c i \u5929\u6211\u4eec\u9009\u62e9\u4e86\u7b2c\u4e09\u79cd\uff0c\u90a3\u6b64\u65f6 dp[ i ] = max\uff08\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e00\u79cd\uff0c\u4e0a\u4e00\u5929\u9009\u62e9\u7b2c\u4e8c\u79cd\uff09 + c[ i ]\u3002<\/p>\n\n\n\n<p>\u6b64\u65f6\u6211\u4eec\u53ef\u4ee5\u5199\u51fa\u4e09\u79cd\u9012\u63a8\u5f0f\uff1a<\/p>\n\n\n\n<p>$dp_1[i]=max(dp_2[i &#8211; 1],dp_2[i &#8211; 1])+a[i]$<\/p>\n\n\n\n<p>$dp_2[i]=max(dp_1[i &#8211; 1],dp_3[i &#8211; 1])+b[i]$<\/p>\n\n\n\n<p>$dp_3[i]=max(dp_1[i &#8211; 1],dp_2[i &#8211; 1])+c[i]$<\/p>\n\n\n\n<p>\u6700\u540e\u6211\u4eec\u4e09\u79cd\u53d6max\u5c31\u884c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 1e5 + 7;\nint a&#91;MAXN];\nint b&#91;MAXN];\nint c&#91;MAXN];\nint dp1&#91;MAXN];\nint dp2&#91;MAXN];\nint dp3&#91;MAXN];\nint main(){\n\tint n;\n\tcin &gt;&gt; n;\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tcin &gt;&gt; a&#91;i] &gt;&gt; b&#91;i] &gt;&gt; c&#91;i];\n\t}\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tdp1&#91;i] = max(dp2&#91;i - 1],dp3&#91;i - 1]) + a&#91;i];\n\t\tdp2&#91;i] = max(dp1&#91;i - 1],dp3&#91;i - 1]) + b&#91;i];\n\t\tdp3&#91;i] = max(dp1&#91;i - 1],dp2&#91;i - 1]) + c&#91;i];\n\t}\n\tcout &lt;&lt; max(max(dp1&#91;n],dp2&#91;n]),dp3&#91;n]) &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">C &#8211; Knapsack 2<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Problem Statement<\/h3>\n\n\n\n<p>Peter\u73b0\u6709 <strong>N<\/strong> \u4e2a\u7269\u54c1\uff0c\u5e8f\u53f7\u5206\u522b\u4e3a <strong>1, 2, &#8230; , N<\/strong>\u3002\u5bf9\u4e8e\u6bcf\u4e2a <strong>i<\/strong> (<strong>1 \u2264 i \u2264 N<\/strong>)\uff0c\u7269\u54c1 <strong>i<\/strong> \u6709\u4e00\u4e2a\u4f53\u79ef <strong>wi<\/strong> \u548c\u4e00\u4e2a\u4ef7\u503c <strong>vi<\/strong>\u3002<\/p>\n\n\n\n<p>Peter\u60f3\u5728\u8fd9 <strong>N<\/strong> \u4e2a\u7269\u54c1\u4e2d\u9009\u53d6\u4e00\u4e9b\u653e\u5230\u5927\u80cc\u5305\u91cc\u5e26\u56de\u5bb6\u3002\u5df2\u77e5\u80cc\u5305\u7684\u5bb9\u79ef\u4e3a <strong>W<\/strong>\uff0c\u8fd9\u610f\u5473\u7740\u6240\u5e26\u7269\u54c1\u7684\u603b\u4f53\u79ef\u4e0d\u80fd\u8d85\u8fc7 <strong>W<\/strong>\u3002<\/p>\n\n\n\n<p>\u6c42\u51faPeter\u53ef\u4ee5\u5e26\u56de\u5bb6\u7684\u7269\u54c1\u603b\u4ef7\u503c\u53ef\u80fd\u7684\u6700\u5927\u503c\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Constraints<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8f93\u5165\u7684\u6240\u6709\u6570\u503c\u5747\u4e3a\u6574\u6570\u3002<\/li>\n\n\n\n<li>$1 \u2264 N \u2264 100$<\/li>\n\n\n\n<li>$1 \u2264 W \u2264 10^9$<\/li>\n\n\n\n<li>$1 \u2264 wi \u2264 W$<\/li>\n\n\n\n<li>$1 \u2264 vi \u2264 10^3$<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Input<\/h3>\n\n\n\n<p>\u6807\u51c6\u8f93\u5165\u683c\u5f0f\u5982\u4e0b\uff1aN W<br>w1 v1<br>w2 v2<br>:<br>wN vN<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Output<\/h3>\n\n\n\n<p>\u8f93\u51faPeter\u53ef\u4ee5\u5e26\u56de\u5bb6\u7684\u7269\u54c1\u603b\u4ef7\u503c\u53ef\u80fd\u7684\u6700\u5927\u503c\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/C\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u8fd9\u5c31\u662f\u5f88\u7ecf\u5178\u7684\u80cc\u5305DP\u3002\u6211\u4eec\u50cf\u641c\u7d22\u4e00\u6837\u6765\u601d\u8003\u8fd9\u4e48\u4e24\u4e2a\u95ee\u9898\uff1a\u600e\u4e48\u5b9a\u4e49\u80cc\u5305\u7684\u72b6\u6001\uff0c\u600e\u4e48\u5b9a\u4e49\u8f6c\u79fb\u3002<\/p>\n\n\n\n<p>\u9996\u5148\uff0c\u600e\u4e48\u5b9a\u4e49\u80cc\u5305\u7684\u72b6\u6001\uff0c\u6211\u4eec\u5c31\u6765\u601d\u8003\u6709\u54ea\u4e9b\u6570\u636e\u9700\u8981\u88ab\u8bb0\u5f55\uff1a\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u4ef7\u503c\u3001\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u91cd\u91cf\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5f00\u4e2a\u4e00\u7ef4\u6570\u7ec4\u5904\u7406\u8fd9\u4ef6\u4e8b\uff1a\u5047\u8bbe\u7528dp [ i ] \u8868\u793a\u3010\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u4ef7\u503c\u3011\uff0c\u7528 i \u8868\u793a\u3010\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u91cd\u91cf\u3011\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u6211\u4eec\u5047\u8bbe\u6b64\u65f6\u7269\u54c1\u7684\u603b\u91cd\u91cf\u662f i ,\u4e0b\u4e00\u4e2a\u653e\u8fdb\u6765\u7684\u7269\u54c1\u7684\u4e0b\u6807\u662f j ,\u5c31\u5f97\u5230\uff1a<\/p>\n\n\n\n<p>$\\text{dp}[i + w_j] = \\text{dp}[i] + v_j$<\/p>\n\n\n\n<p>\u7136\u540e\u6211\u4eec\u4e5f\u4e0d\u77e5\u9053 i \u548c j \u662f\u8c01\uff0c\u628a j \u4ece1 \u5230 n \u679a\u4e3e\u4e00\u4e0b\uff0c\u628a i \u4ece 0 \u5230 $W &#8211; W[j]$ (\u80fd\u8ba9\u7269\u54c1 j \u8fdb\u6765\u7684\u7a7a\u95f4\uff09\uff0c\u6700\u540e\u53d6max\u503c\u5c31\u597d\u3002<\/p>\n\n\n\n<p>\u4f46\u662f\u8981\u6ce8\u610f\u4e00\u70b9\uff0c\u56e0\u4e3a\u653e\u8fdb\u6765\u7684\u7269\u54c1\u4f1a\u5f71\u54cd\u5230\u4e4b\u540e\u7684\u5224\u65ad\uff0c\u5373\u6211\u4eec\u4f1a\u628a\u7269\u54c1 j \u591a\u6b21\u653e\u5165\u80cc\u5305\u3002<\/p>\n\n\n\n<p>\u8fd9\u662f\u4e3a\u4ec0\u4e48\uff0c\u56e0\u4e3a\u6211\u4eec\u6c42 $dp[ i + w_j]$ \u7684\u72b6\u6001\u7684\u65f6\u5019\uff0c\u662f\u4f9d\u636e $dp[i]$ \u7684\u72b6\u6001\u518d\u52a0\u4e0a j \u7269\u54c1\u7684\u4ef7\u503c<\/p>\n\n\n\n<p>\u4f46\u662f\u6211\u4eec\u4e0d\u77e5\u9053dp[i]\u91cc\u9762\u662f\u5426\u5df2\u7ecf\u5305\u542b j \u7269\u54c1\u4e86\uff0c\u56e0\u4e3a j \u6bcf\u6b21\u90fd\u4ece 1 \u5f00\u59cb\u679a\u4e3e\u5230 n (\u5373\u6240\u6709\u7269\u54c1)<\/p>\n\n\n\n<p>\u6240\u4ee5\u6211\u4eec\u5e94\u8be5\u53cd\u5411\u679a\u4e3e i \uff0c\u5373\u4eceW\u679a\u4e3e\u5230$W_j$\uff0c\u8f6c\u6362\u5f0f\u53d8\u4e3a $\\text{dp}[i] = \\text{dp}[i &#8211; w_j] + v_j$ \uff0c\u6b64\u65f6\u524d\u9762\u8f83\u5c0f\u7684 $dp[i]$ \u5c31\u4e0d\u4f1a\u5f71\u54cd\u5230\u540e\u9762\u8f83\u5927\u7684 $dp[ i ]$ \uff0c\u56e0\u4e3a\u6211\u4eec\u5e95\u5c42\u7684\u601d\u60f3\u6ca1\u6709\u53d8\uff0c\u6240\u4ee5\u662f\u53ef\u4ee5\u7406\u89e3\u7684\u3002<\/p>\n\n\n\n<p><strong>\u4f46\u662f\uff01\u5bf9\u4e8e\u8fd9\u9053\u9898\u76ee\uff01<\/strong><\/p>\n\n\n\n<p>\u8fd9\u4e2a\u65f6\u5019\u6211\u4eec\u6765\u770b\u8fd9\u4e2a\u6570\u636e\u8303\u56f4\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>$1 \u2264 N \u2264 100$<\/li>\n\n\n\n<li>$1 \u2264 W \u2264 10^9$<\/li>\n\n\n\n<li>$1 \u2264 wi \u2264 W$<\/li>\n\n\n\n<li>$1 \u2264 vi \u2264 10^3$<\/li>\n<\/ul>\n\n\n\n<p>\u6211\u4eec\u53ef\u4ee5\u770b\u5230\uff0cW\u7684\u6570\u636e\u8303\u56f4\u9ad8\u8fbe1e9\uff0c\u6839\u672c\u6ca1\u529e\u6cd5\u5f00\u6570\u7ec4\u505a\uff0c\u6240\u4ee5\u6211\u4eec\u5f97\u6362\u6362\u601d\u8def\u3002\u6211\u4eec\u53ef\u4ee5\u770b\u5230V\u7684\u6570\u636e\u8303\u56f4\u5c0f\uff0c\u6211\u4eec\u53ef\u4ee5\u53cd\u8fc7\u6765\u505a\u5904\u7406\uff0c\u7528 dp [ i ] \u8868\u793a\u3010\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u91cd\u91cf\u3011\uff0c\u7528 i \u8868\u793a\u3010\u80cc\u5305\u4e2d\u7269\u54c1\u7684\u4ef7\u503c\u3011\u3002\u6700\u540e\u904d\u5386\u4e00\u904ddp\u6570\u7ec4\uff0c\u5c31\u53ef\u4ee5\u627e\u5230\u6700\u5927\u7684\u6709\u6548 i \u4e86\uff0c\u6b64\u65f6 i \u7684\u8303\u56f4\u57281e5\u5de6\u53f3\uff0c\u662f\u53ef\u4ee5\u5b9e\u73b0\u7684\u3002<\/p>\n\n\n\n<p>\u91cd\u65b0\u601d\u8003\u4e00\u4e0b\u8f6c\u79fb\uff0c\u6211\u4eec\u5047\u8bbe\u6b64\u65f6\u7269\u54c1\u7684\u603b\u4ef7\u503c\u662f i ,\u4e0a\u4e00\u4e2a\u653e\u8fdb\u6765\u7684\u7269\u54c1\u7684\u4e0b\u6807\u662f j \uff0c<\/p>\n\n\n\n<p>\u5c31\u5f97\u5230\uff1a $dp[i] = dp[i &#8211; v_j] + w[j]$ ,\u6211\u4eec\u4f9d\u7136\u66b4\u529b\u679a\u4e3e i \u548c j \u5c31\u884c\uff0c\u5e76\u4e14\u4f9d\u7136\u9075\u5faa\u4e4b\u524d\u8bf4\u7684 i \u8981\u53cd\u5411\u904d\u5386\uff0c\u4ece \u6700\u5927\u53ef\u80fd\u7684\u4ef7\u503c100000\u904d\u5386\u5230 $vv[ j ]$ \u3002<\/p>\n\n\n\n<p><strong>\u6ce8\u610f\uff01<\/strong>\u6211\u4eec\u6b64\u65f6\u7684 dp[ i ] \u4ee3\u8868\u7684\u662f\u91cd\u91cf\uff0c\u800c\u4e0d\u662f\u4ef7\u503c\uff0c\u6240\u4ee5\u6211\u4eec\u8981\u7684\u662f\u4e0d\u5927\u4e8eW\u7684\u6700\u5927\u4ef7\u503c\uff0c\u6bcf\u6b21\u8981\u53d6\u7684\u662fmin\u3002\u6240\u4ee5\u5148\u628adp\u6570\u7ec4\u8bbe\u4e3a\u65e0\u7a77\u5927\u5373\u53ef\u3002\u4f46\u662f dp[0] \u8981\u8bbe\u4e3a 0 \uff0c\u5426\u5219\u4f1a\u5f71\u54cd\u9012\u63a8\u7684\u7ed3\u679c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801 0-1\u80cc\u5305\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 1e3 + 7;\nint dp&#91;MAXN];\nint ww&#91;MAXN];\nint vv&#91;MAXN];\n\nint main(){\n\tint n,w;\n\tcin &gt;&gt; n &gt;&gt; w;\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tcin &gt;&gt; ww&#91;i] &gt;&gt; vv&#91;i];\n\t}\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tfor(int j = w;j &gt;= ww&#91;i];j--){\n\t\t\tdp&#91;j] = max(dp&#91;j - ww&#91;i]] + vv&#91;i],dp&#91;j]);\n\t\t}\n\t}\n\n\tcout &lt;&lt; dp&#91;n] &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff08AC)\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 1e5 + 7;\nint dp&#91;MAXN];\nint ww&#91;MAXN];\nint vv&#91;MAXN];\n\nint main(){\n\tint n,w;\n\tcin &gt;&gt; n &gt;&gt; w;\n\tmemset(dp,0x3f,sizeof(dp));\n\tdp&#91;0] = 0;\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tcin &gt;&gt; ww&#91;i] &gt;&gt; vv&#91;i];\n\t}\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tfor(int j = 100000;j &gt;= vv&#91;i];j--){\n\t\t\tdp&#91;j] = min(dp&#91;j],dp&#91;j - vv&#91;i]] + ww&#91;i]);\n\t\t}\n\t}\n\tfor(int i = 100000;i &gt;= 0;i--){\n\t\tif(dp&#91;i] &lt;= w){\n\t\t\tcout &lt;&lt; i &lt;&lt; \"\\n\";\n\t\t\tbreak;\n\t\t}\n\t}\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">D &#8211; LCS<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Problem Statement<\/h3>\n\n\n\n<p>\u7ed9\u5b9a\u5b57\u7b26\u4e32 s \u548c t\u3002\u627e\u5230\u4e00\u4e2a\u6700\u957f\u7684\u5b57\u7b26\u4e32\uff0c\u5b83\u662f s \u548c t \u7684\u5b50\u5e8f\u5217\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Notes<\/h3>\n\n\n\n<p>\u5b57\u7b26\u4e32 x \u7684\u5b50\u5e8f\u5217\u662f\u901a\u8fc7\u5728\u4e0d\u66f4\u6539\u987a\u5e8f\u7684\u60c5\u51b5\u4e0b\u4ece x \u4e2d\u5220\u9664\u96f6\u4e2a\u6216\u591a\u4e2a\u5b57\u7b26\u5e76\u8fde\u63a5\u5176\u4f59\u5b57\u7b26\u800c\u83b7\u5f97\u7684\u5b57\u7b26\u4e32\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Constraints<\/h3>\n\n\n\n<p>s \u548c t \u662f\u7531\u5c0f\u5199\u82f1\u6587\u5b57\u6bcd\u7ec4\u6210\u7684\u5b57\u7b26\u4e32\u3002<\/p>\n\n\n\n<p>$1 \\leq |s|, |t| \\leq 3000$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Input<\/h3>\n\n\n\n<p>\u8f93\u5165\u4ece\u6807\u51c6\u8f93\u5165\u4e2d\u6309\u4ee5\u4e0b\u683c\u5f0f\u7ed9\u51fa\uff1a<\/p>\n\n\n\n<p>$s$<br>$t$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Output<\/h3>\n\n\n\n<p>\u6253\u5370\u4e00\u4e2a\u6700\u957f\u7684\u5b57\u7b26\u4e32\uff0c\u8be5\u5b57\u7b26\u4e32\u662f s \u548c t \u7684\u5b50\u5e8f\u5217\u3002\u5982\u679c\u6709\u591a\u4e2a\u8fd9\u6837\u7684\u5b57\u7b26\u4e32\uff0c\u5219\u5176\u4e2d\u4efb\u4f55\u4e00\u4e2a\u90fd\u5c06\u88ab\u63a5\u53d7\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/D\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge &#8212; NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u6765\u601d\u8003\u4e00\u4e0b\u8fd9\u79cd\u9898\u76ee\u7684\u601d\u8def\u5e94\u8be5\u662f\u4ec0\u4e48\u3002\u7ed9\u4f60\u4e24\u4e2a\u5b57\u7b26\u4e32\uff0c\u53bb\u627e\u6700\u957f\u7684\u516c\u5171\u5b50\u5e8f\u5217\u3002\u5148\u601d\u8003\uff0c\u6211\u4eec\u5e94\u8be5\u600e\u4e48\u5b9a\u4e49\u72b6\u6001\u548c\u8f6c\u79fb\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u9700\u8981\u8bb0\u5f55\u7684\u72b6\u6001\u6709\u54ea\u4e9b\u4e1c\u897f\uff1f\u6211\u4eec\u9700\u8981\u8bb0\u5f55\u6211\u4eec\u904d\u5386\u5230\u4e24\u4e2a\u5b57\u7b26\u4e32\u7684\u54ea\u4e2a\u4f4d\u7f6e\uff0c\u6b64\u65f6\u5177\u6709\u7684\u5b57\u7b26\u4e32\u53c8\u662f\u591a\u957f\u7684\uff0c\u53c8\u662f\u54ea\u4e9b\u5b57\u7b26\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5f00\u4e2a\u4e8c\u7ef4\u6570\u7ec4 $dp[ i ][ j ]$ \uff0c\u7528 i \u548c j \u8bb0\u5f55\u904d\u5386\u5230\u7684\u4f4d\u7f6e\uff0cdp\u7684\u503c\u8868\u793a\u5b57\u7b26\u4e32\u957f\u5ea6\uff0c\u518d\u5f00\u4e00\u4e2a\u5b57\u7b26\u4e32\u53d8\u91cf\u5b58\u50a8\u6700\u957f\u7684\u7ed3\u679c\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u6765\u601d\u8003\u4e00\u4e0b\u8f6c\u79fb\u7684\u8fc7\u7a0b\u3002\u5047\u8bbe\u6211\u4eec\u6b64\u65f6\u904d\u5386\u5230 \u5b57\u7b26\u4e32a\u7684 \u7b2c i \u4f4d \u548c \u5b57\u7b26\u4e32b\u7684 \u7b2c j \u4f4d\uff0c\u5373 $dp[ i ][ j ]$<\/p>\n\n\n\n<p>\u5047\u8bbe\u6b64\u65f6 a[i] == b[j]\uff0c\u90a3\u5c31\u8bf4\u660e\u6211\u4eec\u7684\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u53ef\u5728\u4e4b\u524d\u7684\u57fa\u7840\u4e0a+1\uff0c\u6240\u4ee5 $dp[ i ][ j ] = dp[i &#8211; 1][j &#8211; 1] + 1$<\/p>\n\n\n\n<p>\u53cd\u4e4b\uff0c\u6211\u4eec\u7684\u516c\u5171\u5b50\u5e8f\u5217\u7684\u957f\u5ea6\u5c31\u6ca1\u6709\u53d8\u5316\uff0c$dp[i][j]$ \u7684\u503c\u5c31\u548c $max(dp[i][j &#8211; 1],dp[i &#8211; 1][j])$ \u662f\u4e00\u6837\u7684\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u901a\u8fc7\u56de\u6eaf\u52a8\u6001\u89c4\u5212\u6570\u7ec4 dp\uff0c\u4fbf\u80fd\u6784\u5efa\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u7684\u5b57\u7b26\u4e32 str\u3002<\/p>\n\n\n\n<p>\u56de\u6eaf\u7684\u8fc7\u7a0b\u662f\u8fd9\u6837\u7684\uff1a<\/p>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u5982\u679c s[i &#8211; 1] \u7b49\u4e8e t[j &#8211; 1]\uff0c\u8bf4\u660e\u5f53\u524d\u5b57\u7b26\u662f\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u7684\u4e00\u90e8\u5206\uff0c\u5c06\u8be5\u5b57\u7b26\u52a0\u5165\u5230\u7ed3\u679c\u5b57\u7b26\u4e32 str \u4e2d\uff0c\u5e76\u540c\u65f6\u5c06 i \u548c j \u5411\u524d\u79fb\u52a8\u4e00\u4e2a\u4f4d\u7f6e\u3002<\/li>\n\n\n\n<li>\u5982\u679c s[i &#8211; 1] \u4e0d\u7b49\u4e8e t[j &#8211; 1]\uff0c\u6839\u636e\u52a8\u6001\u89c4\u5212\u6570\u7ec4 dp \u7684\u503c\uff0c\u51b3\u5b9a\u600e\u4e48\u5411\u5de6\u79fb\u52a8\u3002\u5982\u679c $dp[i &#8211; 1][j] &lt; dp[i][j &#8211; 1]$\uff0c\u610f\u5473\u7740\u5411\u5de6\u79fb\u52a8 j \u7684\u503c\u66f4\u6709\u53ef\u80fd\u4f7f\u5f97\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u53d8\u957f\uff0c\u5373 j&#8211; \u3002\u5982\u679c $dp[i &#8211; 1][j] &gt;= dp[i][j &#8211; 1]$\uff0c\u610f\u5473\u7740\u5411\u5de6\u79fb\u52a8 i \u7684\u503c\u66f4\u6709\u53ef\u80fd\u4f7f\u5f97\u6700\u957f\u516c\u5171\u5b50\u5e8f\u5217\u53d8\u957f\uff0c\u5373 i&#8211;\u3002<\/li>\n<\/ul>\n\n\n\n<p>\u6700\u540e\u9006\u5e8f\u8f93\u51fa\u5b57\u7b26\u4e32 str \uff0c\u5373\u53ef\u5f97\u5230\u7ed3\u679c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 3e3 + 7;\nint dp&#91;MAXN]&#91;MAXN];\n\nint main(){\n\tstring s,t;\n\tcin &gt;&gt; s &gt;&gt; t;\n\tint len1 = s.size();\n\tint len2 = t.size();\n\tfor(int i = 1;i &lt;= len1;i++){\n\t\tfor(int j = 1;j &lt;= len2;j++){\n\t\t\tif(s&#91;i - 1] == t&#91;j - 1]) dp&#91;i]&#91;j] = dp&#91;i - 1]&#91;j - 1] + 1;\n\t\t\telse dp&#91;i]&#91;j] = max(dp&#91;i - 1]&#91;j],dp&#91;i]&#91;j - 1]);\n\t\t}\n\t}\n\tstring str = \"\";\n\tint i = len1,j = len2;\n\twhile(i != 0 &amp;&amp; j != 0){\n\t\tif(s&#91;i - 1] == t&#91;j - 1]){\n\t\t\tstr += s&#91;i - 1];\n\t\t\ti--;\n\t\t\tj--;\n\t\t}else{\n\t\t\tdp&#91;i - 1]&#91;j] &lt; dp&#91;i]&#91;j - 1] ? j--:i--;\n\t\t}\n\t}\n\tfor(int i = str.size() - 1;i &gt;= 0;i--)\n\t\tcout &lt;&lt; str&#91;i];\n\tcout &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">E &#8211; Grid 1<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Problem Statement<\/h3>\n\n\n\n<p>\u6709\u4e00\u4e2a\u7f51\u683c\uff0c\u5176\u4e2d\u6709 H \u4e2a\u6c34\u5e73\u884c\u548c W \u4e2a\u5782\u76f4\u5217\u3002\u7528 $(i,j)$ \u8868\u793a\u4ece\u9876\u90e8\u7b2c i \u884c\u548c\u4ece\u5de6\u4fa7\u7b2c j \u5217\u7684\u65b9\u5757\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6bcf\u4e2a i \u548c j $(1 \\leq i \\leq H, 1 \\leq j \\leq W)$\uff0c\u65b9\u5757 $(i,j)$ \u7531\u5b57\u7b26 $a_{i,j}$ \u63cf\u8ff0\u3002\u5982\u679c $a_{i,j}$ \u662f<code>.<\/code>\uff0c\u5219\u65b9\u5757 $(i,j)$ \u662f\u4e00\u4e2a\u7a7a\u65b9\u5757\uff1b\u5982\u679c $a_{i,j}$ \u662f<code>#<\/code>\uff0c\u5219\u65b9\u5757 $(i,j)$ \u662f\u4e00\u4e2a\u5899\u65b9\u5757\u3002\u4fdd\u8bc1\u65b9\u5757 $(1,1)$ \u548c $(H,W)$ \u662f\u7a7a\u65b9\u5757\u3002<\/p>\n\n\n\n<p>Taro \u5c06\u4ece\u65b9\u5757 $(1,1)$ \u5f00\u59cb\uff0c\u901a\u8fc7\u91cd\u590d\u5411\u53f3\u6216\u5411\u4e0b\u79fb\u52a8\u5230\u76f8\u90bb\u7684\u7a7a\u65b9\u5757\uff0c\u5230\u8fbe $(H,W)$\u3002<\/p>\n\n\n\n<p>\u627e\u5230\u4ece\u65b9\u5757 $(1,1)$ \u5230 $(H,W)$ \u7684 Taro \u8def\u5f84\u6570\u91cf\u3002\u7531\u4e8e\u7b54\u6848\u53ef\u80fd\u975e\u5e38\u5927\uff0c\u627e\u5230\u5bf9 $10^9 + 7$ \u53d6\u6a21\u7684\u8ba1\u6570\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Constraints<\/h3>\n\n\n\n<p>H \u548c W \u4e3a\u6574\u6570\u3002 $2 \\leq H, W \\leq 1000$ $a_{i,j}$ \u4e3a<code>.<\/code>\u6216<code>#<\/code>\u3002 \u65b9\u5757 $(1,1)$ \u548c $(H,W)$ \u662f\u7a7a\u65b9\u5757\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Input<\/h3>\n\n\n\n<p>\u8f93\u5165\u4ee5\u4ee5\u4e0b\u683c\u5f0f\u4ece\u6807\u51c6\u8f93\u5165\u7ed9\u51fa\uff1a<\/p>\n\n\n\n<p>$H \\space W$<\/p>\n\n\n\n<p>$a_{1,1} \\ldots a_{1,W}$ <\/p>\n\n\n\n<p>$\\vdots$ <\/p>\n\n\n\n<p>$a_{H,1} \\ldots a_{H,W}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Output<\/h3>\n\n\n\n<p>\u6253\u5370\u4ece\u65b9\u5757 $(1,1)$ \u5230 $(H,W)$ \u7684 Taro \u8def\u5f84\u6570\u91cf\uff0c\u5bf9 $10^9 + 7$ \u53d6\u6a21\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/E\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u6211\u4eec\u8fd8\u662f\u601d\u8003\u8fd9\u4e24\u4e2a\u4e8b\u60c5\uff1a\u72b6\u6001\u548c\u8f6c\u79fb\u3002<\/p>\n\n\n\n<p>\u72b6\u6001\u5c31\u662f\u6211\u4eec\u8981\u8bb0\u5f55\u8def\u5f84\u7684\u4e2a\u6570\u3002<\/p>\n\n\n\n<p>\u8f6c\u79fb\u7684\u8bdd\uff0c\u6bcf\u4e2a\u683c\u5b50\u53ef\u4ee5\u4ece\u5de6\u8fb9\u548c\u4e0a\u9762\u7684\u683c\u5b50\u8f6c\u79fb\u4e0b\u6765\u3002<\/p>\n\n\n\n<p>\u6240\u4ee5\u6211\u4eec\u5f00\u4e2a\u4e8c\u7ef4dp\u6570\u7ec4\uff0c\u7528\u6765\u5b58\u50a8\u4ece\u8d77\u70b9\u5230\u8fbe\uff08i\uff0cj\uff09\u5904\u7684\u8def\u5f84\u4e2a\u6570\u3002\u521d\u59cb\u5316\u8d77\u70b9$dp[1][1] = 1$\uff0c\u9047\u5230\u5899\u80af\u5b9a\u662f\u6ca1\u6709\u8def\u5f84\u7684\uff0c\u6240\u4ee5\u5899\u5c31\u662f0\u3002\u6700\u540e\u8f93\u51fa$dp[h][w]$\u7684\u503c\u5c31\u884c\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 2e3 + 7;\nconst int MOD = 1e9 + 7;\nint dp&#91;MAXN]&#91;MAXN];\nchar mx&#91;MAXN]&#91;MAXN];\n\nint main(){\n\tint h,w;\n\tcin &gt;&gt; h &gt;&gt; w;\n\tfor(int i = 1;i &lt;= h;i++){\n\t\tfor(int j = 1;j &lt;= w;j++){\n\t\t\tcin &gt;&gt; mx&#91;i]&#91;j];\n\t\t}\n\t}\n\tdp&#91;1]&#91;1] = 1;\n\tfor(int i = 1;i &lt;= h;i++){\n\t\tfor(int j = 1;j &lt;= w;j++){\n\t\t\tif(mx&#91;i]&#91;j] == '#'){\n\t\t\t\tdp&#91;i]&#91;j] = 0;\n\t\t\t\tcontinue;\n\t\t\t}\n\t\t\tif(i &gt; 1){\n\t\t\t\tdp&#91;i]&#91;j] += dp&#91;i - 1]&#91;j];\n\t\t\t\tdp&#91;i]&#91;j] %= MOD;\n\t\t\t}\n\t\t\tif(j &gt; 1){\n\t\t\t\tdp&#91;i]&#91;j] += dp&#91;i]&#91;j - 1];\n\t\t\t\tdp&#91;i]&#91;j] %= MOD;\n\t\t\t}\n\t\t}\n\t}\n\tcout &lt;&lt; dp&#91;h]&#91;w] &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">F &#8211; Matching<\/h1>\n\n\n\n<h2 class=\"wp-block-heading\">\u9898\u76ee\uff1a<\/h2>\n\n\n\n<blockquote class=\"wp-block-quote is-layout-flow wp-block-quote-is-layout-flow\">\n<h3 class=\"wp-block-heading\">Problem Statement<\/h3>\n\n\n\n<p>\u6709N\u4e2a\u7537\u4eba\u548cN\u4e2a\u5973\u4eba\uff0c\u5206\u522b\u7f16\u53f7\u4e3a1,2,&#8230;,N\u3002<\/p>\n\n\n\n<p>\u5bf9\u4e8e\u6bcf\u5bf9i,j$\uff081 \\leq i,j \\leq N\uff09$\uff0c\u7537\u4ebai\u548c\u5973\u4ebaj\u7684\u76f8\u5bb9\u6027\u4ee5\u6574\u6570$a_{i,j}$\u7ed9\u51fa\u3002\u5982\u679c$a_{i,j}=1$\uff0c\u5219\u7537\u4ebai\u548c\u5973\u4ebaj\u662f\u76f8\u5bb9\u7684\uff1b\u5982\u679c$a_{i,j}=0$\uff0c\u5219\u4ed6\u4eec\u4e0d\u76f8\u5bb9\u3002<\/p>\n\n\n\n<p>Taro \u8bd5\u56fe\u7ec4\u6210N\u5bf9\uff0c\u6bcf\u5bf9\u7531\u4e00\u4e2a\u7537\u4eba\u548c\u4e00\u4e2a\u5973\u4eba\u7ec4\u6210\uff0c\u4ed6\u4eec\u4e4b\u95f4\u662f\u76f8\u5bb9\u7684\u3002\u5728\u8fd9\u91cc\uff0c\u6bcf\u4e2a\u7537\u4eba\u548c\u6bcf\u4e2a\u5973\u4eba\u5fc5\u987b\u5c5e\u4e8e\u6070\u597d\u4e00\u4e2a\u914d\u5bf9\u3002<\/p>\n\n\n\n<p>\u627e\u51fa Taro \u53ef\u4ee5\u7ec4\u6210N\u5bf9\u7684\u65b9\u5f0f\u6570\uff0c\u53d6\u6a21$10^9+7$\u3002<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Constraints<\/h3>\n\n\n\n<ul class=\"wp-block-list\">\n<li>\u8f93\u5165\u4e2d\u7684\u6240\u6709\u503c\u5747\u4e3a\u6574\u6570\u3002<\/li>\n\n\n\n<li>$1 \\leq N \\leq 21$<\/li>\n\n\n\n<li>$a_{i,j}$\u4e3a0\u62161\u3002<\/li>\n<\/ul>\n\n\n\n<h3 class=\"wp-block-heading\">Input<\/h3>\n\n\n\n<p>\u8f93\u5165\u4ee5\u4ee5\u4e0b\u683c\u5f0f\u4ece\u6807\u51c6\u8f93\u5165\u7ed9\u51fa\uff1a<\/p>\n\n\n\n<p>N <\/p>\n\n\n\n<p>$a_{1,1} a_{1,2} &#8230; a_{1,N}$<\/p>\n\n\n\n<p>$&#8230; $<\/p>\n\n\n\n<p>$a_{N,1} a_{N,2} &#8230; a_{N,N}$<\/p>\n\n\n\n<h3 class=\"wp-block-heading\">Output<\/h3>\n\n\n\n<p>\u8f93\u51fa Taro \u53ef\u4ee5\u7ec4\u6210N\u5bf9\u7684\u65b9\u5f0f\u6570\uff0c\u53d6\u6a21$10^9+7$\u3002<\/p>\n\n\n\n<p><a href=\"https:\/\/vjudge.net\/contest\/607363#problem\/F\">NENU\u5bd2\u5047DP\u7ec3\u4e60 &#8211; Virtual Judge (vjudge.net)<\/a><\/p>\n<\/blockquote>\n\n\n\n<h2 class=\"wp-block-heading\">\u601d\u8def\uff1a<\/h2>\n\n\n\n<p>\u6211\u4eec\u8fd8\u662f\u601d\u8003\u600e\u4e48\u5b9a\u4e49\u72b6\u6001\u8fd8\u6709\u8f6c\u79fb\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u8981\u8bb0\u5f55\u7684\u72b6\u6001\u6709\u54ea\u4e9b\u4fe1\u606f\u5462\uff1f\u6211\u4eec\u4f1a\u53d1\u73b0\u8fd9\u9053\u9898\u76ee\uff0c\u6211\u4eec\u8fde\u72b6\u6001\u90fd\u5f88\u96be\u5b9a\u4e49\uff0c\u56e0\u4e3a\u6211\u4eec\u6ca1\u529e\u6cd5\u901a\u8fc7\u7b2c i \u4e2a\u7537\u4eba\u548c\u7b2c j \u4e2a\u5973\u4eba\u8fd9\u79cd\u65b9\u5f0f\uff0c\u5c31\u628a\u4e00\u79cd\u72b6\u6001\u5b9a\u4e49\u4e0b\u6765\u3002<\/p>\n\n\n\n<p>\u6211\u4eec\u8003\u8651\u4e00\u79cd\u65b0\u7684\u89d2\u5ea6\uff0c\u6211\u4eec\u628a\u6240\u6709\u5973\u751f\u770b\u4f5c\u4e00\u4e2a\u96c6\u5408\uff0c\u96c6\u5408\u91cc\u6bcf\u4e2a\u5973\u751f\u6709\u5339\u914d\u548c\u6ca1\u5339\u914d\u4e24\u79cd\u72b6\u6001\uff0c\u6240\u4ee5\u8fd9\u4e2a\u96c6\u5408\u7684\u5927\u5c0f\u4e3a$2^n$<\/p>\n\n\n\n<p>\u7136\u540e\u6211\u4eec\u7528 $dp_{i,s}$ \u8868\u793a\u524d i \u4e2a\u7537\u751f\u548cS\u4e2d\u5973\u751f\u5339\u914d\u7684\u65b9\u6848\u6570\u3002<\/p>\n\n\n\n<p>\u5219\u53ef\u5f97\u5230\u72b6\u6001\u8f6c\u79fb\uff1a$dp_{i,S} = \\sum{dp_{i-1,S-j},j \\in S \\ \\text{and} \\ g[i][j]=1}$ \uff0c\u5373\u7531\u524d i &#8211; 1\u4e2a\u7537\u751f\u548c { S &#8211; j } \u4e2a\u5973\u751f \u7684\u72b6\u6001\u8f6c\u79fb\u800c\u6765\uff0c\u6b64\u65f6\u7b2c i \u4e2a\u7537\u751f\u8981\u548c\u9009\u4e2d\u7684\u7b2c j \u4e2a\u5973\u751f\u5339\u914d\uff0c\u524d i &#8211; 1 \u4e2a\u7537\u751f\u4e0d\u80fd\u548c\u7b2c j \u4e2a\u5973\u751f\u5339\u914d\u3002\u5982\u679c\u6211\u4eec\u5f00\u4e2a21\u7ef4\u7684 $dp[2][2]&#8230;[2]$ \u4e5f\u592a\u9ebb\u70e6\u4e86\uff0c\u6240\u4ee5\u6211\u4eec\u53ef\u4ee5\u5f00\u4e2a\u6570\u7ec4 $dp[1 &lt;&lt; n]$ \u3002<\/p>\n\n\n\n<p>\u8fd9\u5c31\u662f\u72b6\u538bDP\uff0c\u5f53\u770b\u5230 $1 \\leq N \\leq 21$ \u7684\u65f6\u5019\uff0c\u5176\u5b9e\u5c31\u53ef\u4ee5\u5f80\u72b6\u538b\u7684\u89d2\u5ea6\u601d\u8003\uff0c\u56e0\u4e3a $2^n$ \u4e0d\u662f\u5f88\u5927\u3002<\/p>\n\n\n\n<p>\u4ee3\u7801\u548c\u6ce8\u91ca\u5df2\u7ecf\u5c3d\u91cf\u8bb2\u660e\u767d\u4e86\uff0c\u770b\u4ee3\u7801\u7406\u89e3\u5427\u3002<\/p>\n\n\n\n<p>\u56e0\u4e3a $dp[i][s]$ \u53ea\u548c $dp[i &#8211; 1][s &#8211; j]$ \u6709\u5173\uff0c\u6240\u4ee5\u4e5f\u53ef\u4ee5\u964d\u7ef4\u6210\u4e00\u7ef4\u7684\u5904\u7406\uff0c\u8fd9\u4e2a\u4f18\u5316\u4ea4\u7ed9\u8bfb\u8005\u3002<\/p>\n\n\n\n<h2 class=\"wp-block-heading\">\u4ee3\u7801\uff1a<\/h2>\n\n\n\n<pre class=\"wp-block-code\"><code>#include&lt;bits\/stdc++.h&gt;\nusing namespace std;\nconst int MAXN = 25;\nconst int MOD = 1e9 + 7;\nint dp&#91;22]&#91;4194304];\/\/dp&#91;i]&#91;j]\u8868\u793a\u679a\u4e3e\u5230\u7b2cj\u4e2a\u72b6\u6001\u65f6\uff0c\u524di\u4e2a\u7537\u751f\u90fd\u6709\u5339\u914d\u7684\u65b9\u6848\u6570\nint a&#91;MAXN]&#91;MAXN];\nint count(int n){\n\tint cnt = 0;\n\twhile(n){\n\t\tif(n &amp; 1){\n\t\t\tcnt++;\n\t\t}\n\t\tn = n &gt;&gt; 1;\n\t}\n\treturn cnt;\n}\nint main(){\n\tint n;\n\tcin &gt;&gt; n;\n\tfor(int i = 1;i &lt;= n;i++){\n\t\tfor(int j = 1;j &lt;= n;j++){\n\t\t\tcin &gt;&gt; a&#91;i]&#91;j];\n\t\t}\n\t}\n\t\/\/\u8868\u793a\u5973\u751f\u5339\u914d\u7684\u72b6\u6001\n\tint m = 1 &lt;&lt; n;\n\t\/\/\u6ca1\u6709\u4eba\u76f8\u4e92\u5339\u914d\u7684\u65b9\u6848\u67091\u79cd\n\tdp&#91;0]&#91;0] = 1;\n\t\/\/\u679a\u4e3e\u524d i \u4e2a\u7537\u751f\u5339\u914d k \u4e2a\u5973\u751f\u7684\u65b9\u6848\u6570\n\tfor(int i = 1;i &lt;= n;i++){\n\t\t\/\/\u679a\u4e3e\u6240\u6709\u7684\u5339\u914d\u72b6\u6001\uff0c\u6b64\u65f6\u72b6\u6001j\u8868\u793a\u6709i\u5bf9\u88ab\u5339\u914d\n\t\tfor(int j = 0;j &lt; m;j++){\n\t\t\t\/\/\u5047\u8bbe\u8fd9\u4e2a\u5339\u914d\u72b6\u6001\u548c\u524d i \u4e2a\u7537\u751f\u5df2\u7ecf\u5339\u914d\u8fd9\u4ef6\u4e8b\u4e0d\u7b26\u5408\u5c31\u8df3\u8fc7\n\t\t\tif(count(j) != i){\n\t\t\t\tcontinue;\n\t\t\t}\n\t\t\t\/\/\u6b64\u65f6\u679a\u4e3e\u5230\u7684S\u96c6\u5408\u72b6\u6001\u4ee3\u8868\u5df2\u7ecf\u6709 i \u5bf9\u4e86\n\t\t\t\/\/\u6b64\u65f6\u679a\u4e3e\u8be5\u72b6\u6001\u7684\u6240\u6709\u5339\u914d\u7684\u5973\u751f\uff0c\u5fc5\u987b\u662f\u7537\u5973\u76f8\u5bb9\u4e14\u5e94\u5f53\u8981\u88ab\u5339\u914d\u7684\n\t\t\tfor(int k = 1;k &lt;= n;k++){\n\t\t\t\t\/\/\u8981\u7b2c i \u4e2a\u7537\u751f\u548c\u7b2c k \u4e2a\u5973\u751f\u76f8\u5bb9\uff0c\u5e76\u4e14\u7b2c k \u4e2a\u5973\u751f\u662f\u6211\u4eec\u9009\u4e2d\u51c6\u5907\u5339\u914d\u7684\n\t\t\t\tif(a&#91;i]&#91;k] &amp;&amp; (j &gt;&gt; (k - 1) &amp; 1)){\n\t\t\t\t\t\/\/\u66f4\u65b0dp&#91;i]&#91;j]\u7684\u503c\uff0c\u52a0\u4e0a\u524d i-1 \u4e2a\u7537\u751f\u548c\u7b2c k \u4e2a\u5973\u751f\u672a\u5339\u914d\u7684\u72b6\u6001\u6570\n\t\t\t\t\t\/\/\u524d i-1 \u4e2a\u7537\u751f\u548c\u7b2c k \u4e2a\u5973\u751f\u672a\u5339\u914d\u7684\u72b6\u6001\u6570\u7528\u5f53\u524d\u679a\u4e3e\u5230\u7684\u72b6\u6001j - (1 &lt;&lt; (k - 1)\uff09\u8868\u793a\n\t\t\t\t\t\/\/\u8fd9\u91cc\u7684(1 &lt;&lt; (k - 1)\uff09\u8868\u793a\u72b6\u6001\u8868\u91cc\uff0c\u7b2ck\u4f4d\u5973\u751f\u4e3a\u5339\u914d\u7684\u72b6\u6001\n\t\t\t\t\t\/\/j - (1 &lt;&lt; (k - 1))\u662f\u5c06\u5f53\u524d\u5339\u914d\u72b6\u6001j\u4e2d\u53bb\u6389\u7b2ck\u4e2a\u5973\u751f\u7684\u5339\u914d\u72b6\u6001\n\t\t\t\t\tdp&#91;i]&#91;j] = (dp&#91;i]&#91;j] + dp&#91;i - 1]&#91;j - (1 &lt;&lt; (k - 1))]) % MOD;\n\t\t\t\t}\n\t\t\t}\n\t\t}\n\t}\n\t\/\/\u8f93\u51fa\u524dn\u4e2a\u7537\u751f\u5bf9\u4e8em\u79cd\u72b6\u6001\u7684\u5339\u914d\u6570\n\tcout &lt;&lt; dp&#91;n]&#91;m - 1] &lt;&lt; \"\\n\";\n}<\/code><\/pre>\n\n\n\n<h1 class=\"wp-block-heading\">\u603b\u7ed3<\/h1>\n\n\n\n<p>\u8fd9\u5957\u9898\u96be\u5ea6\u9002\u4e2d\uff0c\u6700\u540e\u4e00\u9898\u72b6\u538b\u6bd4\u8f83\u96be\u6574\uff0c\u4f46\u524d\u51e0\u9898\u51e0\u4e4e\u90fd\u662f\u53d8\u5f62\u9898\uff0c\u8f83\u6709\u6559\u5b66\u610f\u4e49\u3002<\/p>\n\n\n\n<ol class=\"wp-block-list\">\n<li>\u7ebf\u6027DP\u8f83\u4e3a\u7b80\u5355\uff0c\u8003\u8651\u601d\u8def\u5c31\u662f\u201c\u5b9a\u4e49\u72b6\u6001\u201d+\u201c\u5b9a\u4e49\u8f6c\u79fb\u201d\uff0c\u53ef\u4ee5\u5148\u8003\u8651\u7528\u641c\u7d22\u7684\u601d\u8def\u53bb\u505a<\/li>\n\n\n\n<li>\u65e0\u8bba\u662fDP\u8fd8\u662f\u641c\u7d22\u95ee\u9898\uff0c\u6211\u4eec\u90fd\u8981\u6ce8\u610f\u6570\u636e\u8303\u56f4\uff01\uff01\uff01\u6570\u636e\u8303\u56f4\u751a\u81f3\u80fd\u7ed9\u4f60\u4e00\u4e9b\u89e3\u9898\u601d\u8def<\/li>\n\n\n\n<li>DP\u65e0\u677f\u9898\uff0c\u5b66\u4f1a\u4e86\u57fa\u672c\u601d\u60f3\u4e4b\u540e\uff0c\u591a\u505a\u9898\u6536\u96c6\u7ecf\u9a8c\u5f88\u91cd\u8981<\/li>\n\n\n\n<li>\u72b6\u6001\u538b\u7f29\u7684\u601d\u60f3\u4e5f\u633a\u91cd\u8981\uff0c\u53ef\u4ee5\u6536\u96c6\u8d77\u6765<\/li>\n<\/ol>\n","protected":false},"excerpt":{"rendered":"<p>A &#8211; Frog 2 \u9898\u76ee\uff1a \u601d\u8def\uff1a \u8fd9\u9053\u9898\u5982\u679c\u7528\u641c\u7d22\u53bb\u505a\u7684\u8bdd\uff0c\u4e00\u4e2a\u662f\u6570\u636e\u91cf\u592a\u5927\uff0cDFS\u5f88\u53ef\u80fd\u4f1a\u7206 [&hellip;]<\/p>\n","protected":false},"author":1,"featured_media":0,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"footnotes":""},"categories":[12,6],"tags":[],"class_list":["post-86","post","type-post","status-publish","format-standard","hentry","category-12","category-6"],"_links":{"self":[{"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/posts\/86","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/comments?post=86"}],"version-history":[{"count":43,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/posts\/86\/revisions"}],"predecessor-version":[{"id":145,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/posts\/86\/revisions\/145"}],"wp:attachment":[{"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/media?parent=86"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/categories?post=86"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.nenuacm.top\/index.php\/wp-json\/wp\/v2\/tags?post=86"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}